Open Access Austin Journal of Surgery Review Article Fibonacci Series, Golden Proportions, and the Human Biology Dharam Persaud-Sharma* and James P O’Leary Florida International University, Herbert Wertheim Abstract College of Medicine Pythagoras, Plato and Euclid’s paved the way for Classical Geometry. *Corresponding author: Dharam Persaud-Sharma, The idea of shapes that can be mathematically defined by equations led to the Florida International University, Herbert Wertheim creation of great structures of modern and ancient civilizations, and milestones College of Medicine, Miami, FL, 33199, USA, Email: in mathematics and science. However, classical geometry fails to explain the [email protected] complexity of non-linear shapes replete in nature such as the curvature of a flower or the wings of a Butterfly. Such non-linearity can be explained by Received: April 01, 2015; Accepted: June 25, 2015; fractal geometry which creates shapes that emulate those found in nature with Published: July 02, 2015 remarkable accuracy. Such phenomenon begs the question of architectural origin for biological existence within the universe. While the concept of a unifying equation of life has yet to be discovered, the Fibonacci sequence may establish an origin for such a development. The observation of the Fibonacci sequence is existent in almost all aspects of life ranging from the leaves of a fern tree, architecture, and even paintings, makes it highly unlikely to be a stochastic phenomenon. Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body. Keywords: Fibonacci; Surgery; Medicine; Anatomy; Golden proportions; Math; Biology Introduction phenomenon known as the Fibonacci sequence and the rule of ‘Golden Proportions’ may serve as a starting point for uncovering the Classical geometry includes the traditional shapes of triangles, methods of a universal architect, should one exist. squares, and rectangles that have been well established by the brilliance of Pythagoras, Plato, and Euclid. These are fundamental Fibonacci and rule of golden proportions shapes that have forged the great structures of ancient and modern Fibonacci sequences: The Fibonacci sequence was first day civilization. However, classical geometry fails to translate into recognized by the Indian Mathematician, Pingala (300-200 B.C.E.) complex non-linear forms that are observed in nature. To explain in his published book called the Chandaśāstra, in which he studied such non-linear shapes, the idea of fractal geometry is proposed which grammar and the combination of long and short sounding vowels states that such fractal shapes/images possess self-similarity and [1,2]. This was originally known as mātrāmeru, although it is now can be of non-integer or non-whole number dimensions. Whereas known as the Gopala-Hemachandra Number in the East, and the classical geometric shapes are defined by equations and mostly whole Fibonacci sequence in the West [1,2]. Leonardo Pisano developed number (integer) values, the shapes of fractal geometry can be created the groundwork for what is now known as “Fibonacci sequences” to by iterations of independent functions. Observing the patterns the Western world during his studies of the Hindu-Arab numerical created by repeating ‘fractal images’ closely emulate those observed system. He published the groundwork of Fibonacci sequences in in nature, thus questioning whether this truly is the mechanism of his book called Liber Abaci (1202) in Italian, which translates into origination of life on the planet. Two important properties of fractals English as the “Book of Calculations”. However, it should be noted include self-similarity and non-integer or non-whole number values. that the actual term “Fibonacci Sequences” was a tributary to The idea of ‘self-identity’ is that its basic pattern or fractal is the same Leonardo Pisano, by French Mathematician, Edouard Lucas in 1877 at all dimensions and that its repetition can theoretically continue [3]. The Fibonacci sequence itself is simple to follow. It proposes that to infinity. Examples of such repeating patterns at microscopic and for the integer sequence starting with 0 or 1, the sequential number is macroscopic scales can be seen in all aspects of nature from the the sum of the two preceding numbers as in Figure 1. florescence of a sunflower, the snow-capped peaks of the Himalayan Mountains, to the bones of the human body. A) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…. ∞ The observation of self-similarity belonging to fractal geometry found in multiple aspects of nature, leads to the question of whether Or such an occurrence is merely stochastic or whether it has a functional B) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… ∞ purpose. While a singular unifying equation to define the creation Figure 1: Illustration of Fibonacci Sequences. A) Fibonacci Series starting and design of life has yet to be determined, an un-coincidental with 1 B) Fibonacci Series starting with 0. Austin J Surg - Volume 2 Issue 5 - 2015 Citation: Persaud-Sharma D and O’Leary JP. Fibonacci Series, Golden Proportions, and the Human Biology. ISSN : 2381-9030 | www.austinpublishinggroup.com Austin J Surg. 2015;2(5): 1066. Persaud-Sharma et al. © All rights are reserved Dharam Persaud-Sharma Austin Publishing Group Table 1: Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (φ) 1.618. FN + 1 FN (FN + 1)/FN 2 1 2 3 2 1.5000 5 3 1.6666 8 5 1.6000 13 8 1.6250 21 13 1.6154 34 21 1.6190 55 34 1.6176 89 55 1.6182 Figure 2: Golden Rectangle and Golden Spiral. 1.6179 144 89 (Nearest approximation of φ) As shown in Figure 1, Fibonacci sequences can either begin with 0 or 1. Although the sequence, originally proposed in Liber Abaci by Leonardo (follows the sequence in A) begins with 1, this sequence essentially reduces to the recurrence relationship of: FN = Fn-1 + Fn-2 Given that the starting numbers of the sequence can be either: A) F1 = 1 F2 = 1 or B) F1 = 0 F2 = 1.In this sequence it can be seen that each additional number after the given conditions of 1 or 0, is a direct result of the sum of the two preceding numbers. For example, in series (A), 1+ 1 = 2, where 2 is the third number in the sequence. Figure 3: Golden Spiral in Nature. To find the third number (F4), simply add F3 = 2 and F2 = 1, which equals 3 (F4­­). Golden proportion, rectangle, and spiral: Another observation that can be made from the series of the Fibonacci numbers includes the rule of golden proportions. In essence, this is an observation that the ratio of any two sequential Fibonacci numbers approximates to the value of 1.618, which is most commonly represented by the Greek Letter Phi (φ). The larger the consecutive numbers in the sequence, the more accurate the approximation of 1.618. A summary of these results illustrating this concept can be seen in Table 1. Furthering this observation, shapes can be created based upon length measurements of the Fibonacci numbers in sequence. For example, creating a rectangle with the values of any of the two successive Fibonacci numbers form what is known as the “Golden Rectangle”. These rectangles can be divided into squares that are equally sided and are of smaller values from the Fibonacci sequence Figure 2. Figure 4: The human hand superimposed on the Golden Rectangle, with approximations to the Golden Spiral [4]. Rectangles created using consecutive numbers from the Fibonacci sequence can be divided into equally sided squares of such numbers Human anatomy and Fibonacci as well. The length of a single side can be divided into smaller values Proportion of human bones: In the human body, there are of the Fibonacci sequence. Connecting the diagonals of each of the instances of the Fibonacci Phi (φ) although it has not been widely squares creates a spiral, known as the Golden Spiral. discussed. In 1973, Dr. William Littler proposed that by making a This image compares the spiral of the human ear, the shell of a clenched fist, Fibonacci’s spiral can be approximated in Figure 4 [4]. nautilus, and the golden spiral. Both the human ear and the shell of a In this paper, Litter proposed that such geometry is observed nautilus approximate the dimensions of the golden spiral. based upon ratio of the lengths of the phalanges, and that the flexor Connecting the corners of each of the squares by an arc reveals a and extensor movement of the primary fingers approximate the spiral pattern. This spiral pattern is known as the ‘golden spiral’. This golden spiral. Such a spiral would have to be created based upon the pattern is seen in several different forms in nature including, but not relationship between the metacarpophalangeal and interphalanges of limited to the human ear and the shell of a nautilus as seen in Figure 3. the digits [4]. However, due a lack of statistical and well documented Submit your Manuscript | www.austinpublishinggroup.com Austin J Surg 2(5): id1066 (2015) - Page - 02 Dharam Persaud-Sharma Austin Publishing Group have attempted to quantitatively characterize the parameters of an aesthetically appealing smile. The most logical starting point is that of the rule of golden proportions. In 1973, Lombardi was the first to officially propose the existence of having proportionate teeth, but dismissed the idea of using the rule of golden proportions to create aesthetic teeth [9]. He did suggest an underlying repeated ratio of the maxillary anterior teeth for aesthetics, but stated the inappropriate use of the rule of golden proportions for such purposes.